Mathematical Representatio ns Module 2 Mathematics Teaching - - PowerPoint PPT Presentation

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Mathematical Representations

Module 2

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The ratio of boys to girls in a class is 5 to 3. After six girls join the class, the number of boys and girls in the class is the same. How many students are in the class now? Post solutions on chart paper.

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Consider Student Work Samples

• What mathematics does each student

know?

• What do we learn about each child based
• n their responses?
• What are the advantages and

disadvantages of each representation?

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Student Work Sample #1

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Student Work Sample #2

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Student Work Sample #3

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Student Work Sample #4

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Student Work Sample #5

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Student Work Sample #6

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If instruction emphasizes only one model or

• ne representational type, then the

mathematical tools that children acquire is limited.

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• When we talk about graphing ratios, would

the graph be continuous or discrete, and why?

• Is graphing a relationship defined by a ratio

different from graphing a proportional relationship?

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Ratios

• Discrete
• Uses additive

and multiplicative reasoning patterns!

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Dora has been asked to make cupcakes for the entire swim team. She knows that when she makes 1, 2, or 3 cupcakes how much icing she needs. How much icing will she need for the whole team of 45 swimmers?

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Proportions:

• Continuous
• Uses multiplicative

reasoning (aka constant of proportionality)!

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The ratio of girls to boys in a freshman class is 4:3. The ratio of girls to boys in a sophomore class is 5:4. If we combine the two classes, what would be the ratio of girls to boys in the group?

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Terry has an appointment with his barber. He decides to ride his bicycle to the barber shop instead of driving. If he rides his bike at 10 miles per hour, then he gets there an hour late for his appointment. However, if he rides his bike at 15 miles an hour, then he gets there an hour early. How far does Terry live from the barber shop?

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A pie graph has four unequal slices representing favorite soft drink preferences for a sample of n people. Eight more people join the sample and three-fourths of them choose Coca-Cola as their favorite soft drink. This result makes the Coca-Cola slice increase to exactly 54% of the total graph. How many people were initially in the sample?

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A survey about favorite desserts reports:

• 12% of the sample prefer cake
• 4% of the sample prefer cookies
• 28% of the sample prefer ice cream
• 40% of the sample prefer chocolate

What is the smallest number of people that could have taken the survey?

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If

• and
• ,

what is the ratio of to ?

Adapted from “Contest Problem Book V” from 1986 AHSME

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60% of the members of a computer club were girls. Later, 15 boys joined the club. With the addition of boys, do you expect the percentage of girls to be larger than 60% or smaller? Is it possible to find the percentage of female members in the computer club after the addition of boys? If so, what is the new percentage of females? If not, explain what other information you would need to determine the percentage of girls in the computer club.

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As the star player of a basketball game stands at the free throw line, the announcer states that he had hit 78% of his free throws that year. The star player misses the first shot and makes the second. Later in the game he is fouled for the second time. As he moves to the free throw line, the announcer states that he had made 76% of his free throws so far that year. Can you determine how many free throws this player had attempted and how many he had made that year?

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• How might we make the different types of

representations accessible to children?

• What mathematical points are important to

be made explicitly in class?

• What other problems might be used to help

children acquire these different representational schemes?